If I am responsible, I can own up to that, but how is the insurance company supposed to know that her damage claims are true? I didn't see any damage.
If you haven't searched here, let's go back to the first principles:
1) As a general matter if your tree (or branch) falls in someone else's yard, you are not responsible for damage or removal as a legal matter (put aside moral issues or "good neighbor" issues.
1a) The exception to the rule is if you were negligent in maintaining the tree--that is, you knew that it presented a high risk and failed to take appropriate remedial measures. Then you can be held responsible for damages.
Having a branch fall off a tree does not mean that the tree is unhealthy. It could mean that, but it could also be simply one branch. If it was obvious the tree (or that branch) was unhealthy then you may be responsible, and if a neighbor called the problem to your attention then it is difficult to deny that you were aware of the problem. This is a fact question and hard to answer based on a short post here--how aware were you of the danger?
Second, assuming that you were negligent in maintaining your tree then she is responsible for proving the actual damages. Before you pay anything, she should provide an accounting of what the damages were, including receipts for any repair costs or a list of what in fact was damaged (e.g., if it broke an outdoor table, what is the replacement cost). Just asserting there were damages is not enough--she's not entitled to some amount of money just because a branch fell in her yard.
If she has filed a claim with her insurance company, and they have taken responsibility for the claim, then there is no reason for you to compensate her as well for any damage. They will make her prove her damages. She is also responsible for her deductible. If the insurance company comes after you, then you should turn their claim over to your insurance company, and let the two of them fight it out. You should cooperate with your insurance company so that they can properly defend your claim and determine whose insurance pays for the damage, hers or yours. If they determine you were responsible then her deductible will be covered by your insurance company or by her, depending on the policy, and that is the extent of your responsibility. Of course, if the insurance companies determine the loss is on her, then she pays her deductible and you are not responsible for covering what her insurance does not.
If she pushes you tell her the basic rule and also that because she has made a claim with her insurance company that you have turned over the claim to your insurance company and the two companies will work it out.